The Future of Democracy
Reflecting on the health of the American experiment and potential reforms.
Key Questions
- Analyze the greatest threat to American democracy today.
- Design strategies to make government more responsive to the needs of the youth.
- Evaluate whether the U.S. Constitution is still an adequate framework for the 21st century.
Common Core State Standards
About This Topic
Introduction to trigonometric ratios, Sine, Cosine, and Tangent, defines the constant relationships between the angles and side lengths of right triangles. In 9th grade, students learn that for any given angle, the ratio of the sides (e.g., opposite over hypotenuse) remains the same regardless of the triangle's size. This is a foundational Common Core standard that connects geometry to the study of periodic functions and physics.
Students learn the SOH-CAH-TOA mnemonic to remember the ratios and explore how these values are stored in calculators. This topic comes alive when students can engage in 'ratio hunts', measuring different-sized triangles with the same angles to discover that the ratios are identical. Collaborative investigations help students see that trigonometry is just a specialized form of similarity that allows us to 'solve' a triangle with very little information.
Active Learning Ideas
Inquiry Circle: The Ratio Discovery
Groups are given several right triangles of different sizes, but all with a 30-degree angle. They must measure the 'opposite' and 'hypotenuse' for each and calculate the ratio. They then compare their results to discover that the ratio is always 0.5, regardless of the triangle's size.
Think-Pair-Share: SOH-CAH-TOA Scenarios
Give students several triangles with one angle and one side marked. Pairs must decide which ratio (Sine, Cosine, or Tangent) is the 'right tool for the job' to find the missing side and explain their choice based on the 'opposite,' 'adjacent,' and 'hypotenuse' labels.
Simulation Game: The Human Trig Table
Students use a large protractor to create different angles on the floor. They measure the 'legs' of the resulting right triangle and calculate the tangent (rise/run). They then compare their 'human' values to the official trig table to see how accurate they were.
Watch Out for These Misconceptions
Common MisconceptionStudents often confuse 'opposite' and 'adjacent' sides, especially when the triangle is rotated.
What to Teach Instead
Use the 'SOH-CAH-TOA Scenarios' activity. Peer teaching helps students realize that 'opposite' is the side 'across' from the angle, while 'adjacent' is the side that 'touches' the angle but isn't the hypotenuse.
Common MisconceptionThinking that trig ratios only work for specific 'special' triangles.
What to Teach Instead
Use 'The Ratio Discovery' activity. Collaborative measurement of 'random' triangles helps students see that these ratios are universal properties of all right triangles, which is why the calculator can store them for every possible angle.
Suggested Methodologies
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Frequently Asked Questions
What does SOH-CAH-TOA stand for?
How can active learning help students understand trig ratios?
Why do we need trigonometry if we have the Pythagorean theorem?
What is the 'Tangent' of an angle?
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